Dimension reduction approach for understanding resource-flow resilience to climate change

The increasing impact of climate change highlights the need for robust methods to assess the resilience of lifeline infrastructures. Traditional dynamical systems approaches often require detailed system descriptions, which can be challenging to obtain for complex socio-technical systems. Network-based approaches offer a valuable alternative, and the dimension-reduction approach, originally developed in ecology, provides a powerful tool to map high-dimensional dynamics to a low-dimensional representation capturing system-level behavior. This approach considers network structure, generating a reduced set of equations describing macroscopic behavior. This study extends this approach to resource-flow networks, specifically applying it to the San Francisco Fuel Transportation Network (SFFTN). By analyzing the impact of sea-level rise (SLR) under various climate change scenarios, the research aims to understand the system's resilience to coastal flooding and production failures, offering insights into the network's ability to maintain fuel supply under stress.

Research on system resilience has intensified due to climate change's increasing effects on lifeline systems. Resilience analysis examines system behavior during failures across preparation, robustness, and recovery phases. Dynamical Systems theory, focusing on parameter changes under different scenarios, is a common approach. However, traditional methods require detailed system dynamics, often unavailable. Network-based approaches address this limitation by modeling system topology and assessing structural changes' impacts. The dimension-reduction approach combines network structure and dynamics, enabling large-scale system dynamics assessment while considering interaction patterns. Initially used in ecology, this approach simplifies complex dynamics to a smaller set of equations representing macroscopic behavior. This method captures network topology's influence on system dynamics through parameters within the reduced equations. Previous work has applied the dimension-reduction method to ecological networks, demonstrating its effectiveness in capturing system-level behaviors and understanding resilience.

The study focuses on the SFFTN, a multiplex network representing fuel transportation from refineries (production) to terminals (intermediate storage) and gas stations (consumption). The network's structure incorporates product pipelines (refineries to terminals) and road networks (terminals and refineries to gas stations). Coastal flooding scenarios, based on various Representative Concentration Pathways (RCPs) and General Circulation Models (GCMs), are used to simulate the impact of climate change on the network's topology. Ordinary differential equations model fuel flow, considering production, consumption, and flow capacities between facilities. These equations capture the qualitative behavior of supply and demand, focusing on system-level behavior rather than individual facility details. A dimension-reduction technique is employed to simplify the original system of over 3400 equations to a manageable set of three equations, representing the average fuel stored in each facility layer (refineries, terminals, gas stations). The approximation assumes low correlation between system parameters (stock, production, demand, flow capacities) and their associated level functions. Aggregated data from the California Energy Commission (CEC) is used to estimate system parameters, considering ranges to reflect uncertainty. The reduced model allows for the analysis of the system's stability under various SLR scenarios and production failures. The model assesses the maximum sustainable demand under different scenarios and the system's transient response to production interruptions, focusing on the time to demand failure (τ) and the average demand level during failure (Q̄).

The dimension-reduction approach reveals that the space of stable macroscopic flows is constrained by the maximum flow theorem between layers, highlighting the network structure's influence on system stability. The SFFTN's ability to sustain demand is assessed under different SLR scenarios. The model predicts that the network can sustain current demand until 2060. However, demand failures are likely after 2060, especially under high greenhouse gas emission scenarios (RCP 8.5). Analysis of production interruptions reveals three potential outcomes: for short interruptions, the system recovers; for intermediate durations, it may reach demand failure even after production resumes; and for long interruptions, all layers are depleted before production is restored. The time to demand failure (τ) is highly dependent on the initial resource level, decreasing under increased coastal flooding. The average demand level during failure (Q̄) decreases linearly with failure duration. Coastal flooding significantly reduces the flow capacities (particularly from refineries to terminals), and the impact on the maximum sustainable demand is more pronounced in later time horizons and higher SLR percentiles. The study's sensitivity analysis confirms the accuracy of the reduced model even under parameter perturbations.

The findings address the research question by demonstrating the dimension-reduction approach's effectiveness in analyzing resource-flow resilience. The results highlight the vulnerability of the SFFTN to SLR and production failures, providing quantitative estimates of the system's capacity under stress. The study's significance lies in bridging traditional network metrics and practical decision-making tools. The approach allows for an analytical tractable representation of complex systems, directly linking network topology changes to the system's capacity as a resource supplier. This contrasts with traditional approaches that often only consider static network characteristics. The integration of climate change scenarios adds a layer of realism, providing valuable insights for resource management and infrastructure planning.

This study successfully demonstrates the application of a dimension-reduction approach to understand the resilience of a complex socio-technical system (the SFFTN) to climate change impacts. The findings highlight the importance of considering both network topology and dynamics in resilience assessments. Future research could extend this approach to other lifeline systems, incorporating more detailed data and exploring interdependencies between different infrastructure networks. Further investigation into different network structures and their influence on the dynamics of networked ordinary differential equations would improve understanding of how topology shapes system behavior.

The study relies on aggregated data, potentially overlooking internal correlations within the network. The model's accuracy depends on the assumption of low correlation between system parameters and level functions. While sensitivity analysis shows robustness, significant correlations could affect the results. The model's simplification may not capture the full complexity of real-world dynamics, and the lack of detailed information on individual facility characteristics limits the precision of the estimates.